Completely convex functions and Lidstone series

Abstract

on an interval is necessarily entire. In that note it was stated that we would show later the relation of this result to the problem of the representation of functions by Lidstone series. It is the aim of the present paper to study this relation. A Lidstone series is a generalization of Taylor's series. It approximates to a given function in the neighborhood of two points instead of one. Such series have been studied by G. J. Lidstone(2), H. Poritsky(3), J. M. Whittaker(4), I. J. Schoenberg(5) and others. More precisely the series has the form